We present a novel spectroscopic technique for second harmonic generation (SHG) using femtosecond laser pulses at 30~kHz repetition rate, which nevertheless provides high spectral resolution limited only by the spectrometer. The potential of this method is demonstrated by applying it to the yellow exciton series of Cu$_2$O. Besides even parity states with $S-$ and $D-$ envelope, we also observe odd parity, $P-$ excitons with linewidths down to 100 $mu$eV, despite of the broad excitation laser spectrum with a full width at half maximum of 14~meV. The underlying light-matter interaction mechanisms of SHG are elaborated by a group theoretical analysis which allows us to determine the linear and circular polarization dependences, in good agreement with experiment.
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